Average Error: 15.1 → 4.8
Time: 7.1s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
↓
\[\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}{\sqrt[3]{z}} \]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}{\sqrt[3]{z}}

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (* x (/ (* (cbrt y) (cbrt y)) (* (cbrt z) (cbrt z))))
  (/ (/ (cbrt y) (fma z z z)) (cbrt z))))
double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}
↓
double code(double x, double y, double z) {
	return (x * ((cbrt(y) * cbrt(y)) / (cbrt(z) * cbrt(z)))) * ((cbrt(y) / fma(z, z, z)) / cbrt(z));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original15.1
Target4.2
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

  1. Initial program 15.1

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

  2. Simplified8.7

    \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z}} \]

  3. Applied add-cube-cbrt_binary649.1

    \[\leadsto x \cdot \frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

  4. Applied *-un-lft-identity_binary649.1

    \[\leadsto x \cdot \frac{\frac{y}{\color{blue}{1 \cdot \mathsf{fma}\left(z, z, z\right)}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

  5. Applied add-cube-cbrt_binary649.2

    \[\leadsto x \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{1 \cdot \mathsf{fma}\left(z, z, z\right)}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

  6. Applied times-frac_binary649.2

    \[\leadsto x \cdot \frac{\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1} \cdot \frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

  7. Applied times-frac_binary649.2

    \[\leadsto x \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}{\sqrt[3]{z}}\right)} \]

  8. Applied associate-*r*_binary644.8

    \[\leadsto \color{blue}{\left(x \cdot \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}{\sqrt[3]{z}}} \]

  9. Simplified4.8

    \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)} \cdot \frac{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}{\sqrt[3]{z}} \]

  10. Final simplification4.8

    \[\leadsto \left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\mathsf{fma}\left(z, z, z\right)}}{\sqrt[3]{z}} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))